Remote Access Journal of the American Mathematical Society
Green Open Access

Journal of the American Mathematical Society

ISSN 1088-6834(online) ISSN 0894-0347(print)

 
 

 

Periods of automorphic forms


Authors: Hervé Jacquet, Erez Lapid and Jonathan Rogawski
Journal: J. Amer. Math. Soc. 12 (1999), 173-240
MSC (1991): Primary 11F55, 11F70, 11F72
DOI: https://doi.org/10.1090/S0894-0347-99-00279-9
MathSciNet review: 1625060
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $E/F$ be a quadratic extension of number fields and $G= \operatorname{Res}_{E/F}H$, where $H$ is a reductive group over $F$. We define the integral (in general, non-convergent) of an automorphic form on $G$ over $H(F)\backslash H(\mathbb A)^1$ via regularization. This regularized integral is used to derive a formula for the integral over $H(F)\backslash H(\mathbb A)^1$ of a truncated Eisenstein series on $G$. More explicit results are obtained in the case $H=GL(n)$. These results will find applications in the expansion of the spectral side of the relative trace formula.


References [Enhancements On Off] (What's this?)

  • [A1] J. Arthur, A trace formula for reductive groups I, Duke Math. J., 45, No. 1 (1978), 911-952. MR 80d:10043
  • [A2] J. Arthur, A trace formula for reductive groups II, Compositio Math., 40 (1980), 87-121. MR 81b:22018
  • [A3] J. Arthur, The trace formula in invariant form, Duke Math. J., 114 (1981), 1-74. MR 84a:10031
  • [A4] J. Arthur, On the inner product of truncated Eisenstein series, Duke Math. J., 49 (1982), 35-70. MR 83e:22023
  • [AS] T. Asai, On certain Dirichlet series associated with Hilbert modular forms and Rankin's method, Math. Ann. 266 (1977), 81-94. MR 55:2761
  • [BZ] I. Bernstein and A. Zelevinsky, Induced representations of reductive ${\mathfrak {p}}$-adic groups I, Ann. scient. Éc. Norm. Sup., 4$^{e}$ série, t. 10, 1977, p. 441-472. MR 58:28310
  • [BR] D. Blasius and J. Rogawski, Cohomology of congruence subgroups of $SU(2,1)^{p}$ and Hodge cycles on some special complex hyperbolic surfaces, to appear in: Proc. of conference on Regulators, Jerusalem 1996 (Ed. A. Resnikov and N. Schappacher).
  • [C] W. Casselman, Extended automorphic forms on the upper half-plane, Math. Annalen 296 (1993), 755-762. MR 94k:11049
  • [F1] Y. Flicker, Twisted Tensor and Euler Products, Bull. Soc. Math. France, 52 (1989), 295-313. MR 89m:11049
  • [F2] Y. Flicker, On distinguished representations, J. Reine Angew. Math. 418 (1991), 139-172. MR 92i:22019
  • [F3] Y. Flicker, Distinguished Representations and a Fourier Trace Formula, Bull. Soc. Math. France, 120 (1992), 413-465. MR 93j:22033
  • [F4] Y. Flicker, Cyclic automorphic forms on a unitary group, J. Math. Kyoto Univ. 37 (1997), 367-439. CMP 98:07
  • [FH] Y. Flicker and J. Hakim, Quaternionic Distinguished Representations, American J. Math., 116 (1994), 255-278. MR 95i:22028
  • [FJ] S. Friedberg and H. Jacquet, Linear periods, J. Reine Angew. Math. 443 (1993), 91-139. MR 94k:11058
  • [FGJR] S. Friedberg, S. Gelbart, H. Jacquet, and J. Rogawski, On genericity for $U(3)$ $L$-packets, pre-print.
  • [GJR] S. Gelbart, H. Jacquet, and J. Rogawski, The relative trace formula and genericity of L-packets for $U(3).$ In preparation.
  • [GRS] S. Gelbart, D. Soudry, and J. Rogawski, Endoscopy, theta-liftings, and period integrals for the unitary group in three variables$.$ Annals of Math. 145 (1997), 1-58. CMP 97:14
  • [GU] J. Guo, On the positivity of central critical values of automorphic L-functions of GL(2), Duke Math. J., 83 (1996), 157-190. MR 97h:11055
  • [G] N. Gurevitch, Unpublished notes.
  • [HLR] G. Harder, R. Langlands, and M. Rapoport, Algebraische zyklen auf Hilbert-Blumenthal Flächen, J. reine angew. Math. 366 (1986), 53-120. MR 87k:11066
  • [J1] H. Jacquet, Sur un résultat de Waldspurger, Ann. Scient. Ec. Norm. Sup., 19 (1986), 185-229. MR 88d:11051
  • [J2] H. Jacquet, The continuous spectrum of the relative trace formula for $GL(3)$ over a quadratic extension, Israel J. Math., 89 (1995), 1-59. MR 96a:22029
  • [J3] H. Jacquet, Automorphic spectrum of symmetric spaces, In: Representation Theory and Automorphic Forms, (Eds. T. N. Bailey and A. W. Knapp) Proc. Symposia in Pure Math. 61, AMS, Providence, 1997. CMP 98:03
  • [JLR] H. Jacquet, K. Lai, S. Rallis, A trace formula for symmetric spaces, Duke Math. J., ${\bf 70}$, No. 2 (1993), 305-372. MR 94d:11033
  • [JL] H. Jacquet and K. Lai, A relative trace formula, Comp. Math., 54 (1985), 243-310. MR 86j:11059
  • [JR] H. Jacquet and S. Rallis, Symplectic periods, J. Reine Angew. Math., 423 (1992), 175-197. MR 93b:22035
  • [JS] H. Jacquet and J. Shalika, On Euler products and the classification of automorphic representations I,II, Amer. J. Math. 103 (1981), 499-558 and 777-815. MR 82m:10050a; MR 82m:10050b
  • [JY] H. Jacquet and Y. Ye, Une remarque sur le changement de base quadratique, C.R. Acad. Sci. Paris Sér. I , Math. 311 (1990), 671-676. MR 92j:11046
  • [JI] D. Jiang, Periods of residual representations of $G_{2},$, pre-print.
  • [LA1] K. F. Lai, Algebraic Cycles on Compact Shimura Surface, Math. Z. 189 (1985), 593-602. MR 87a:11057
  • [LA2] K. F. Lai, On Arthur's Class Expansion of the Relative Trace Formula, Duke Math. J. 64 (1991), 111-117. MR 92k:22031
  • [L] R. Langlands, Eisenstein series, in Algebraic Groups and Discontinuous Subgroups, Proc. Symp. Pure Math. vol. 9, AMS 1966. MR 40:2784
  • [LP] E. Lapid, Thesis, Weizmann Institute of Science, 1998.
  • [LE] J. Levy, A note on the relative trace formula, Canad. Math. Bull. vol. 38 (1995), 450-461. MR 97d:11085
  • [LR] G. Laumon and M. Rapoport, The Langlands lemma and the Betti numbers of stacks of $G$-bundles on a curve, Internat. J. Math. 7 (1996), 29-45. MR 97f:14012
  • [MA] Z. Mao, Relative Kloosterman integrals for the unitary group, C.R. Acad. Sci. Paris Sér. I Math 315 (1992), 381-386. MR 93g:11051
  • [MW] C. Moeglin and J.-L. Waldspurger, Decomposition spectrale et series D'Eisenstein, Birkhauser 1993. MR 95d:11067
  • [SA] P. Sarnak, Diophantine problems and linear groups, In: Proceedings of Int. Cong. Math., Kyoto 1990, Springer Verlag 1991, 459-471. MR 93g:11054
  • [S] T. Springer, Some results on algebraic groups with involutions, in Algebraic groups and related topics, Advanced Studies in Pure Mathematics, vol. 6, 1985, 525-543. MR 86m:20050
  • [Z] D. Zagier, The Rankin-Selberg method for automorphic functions which are not of rapid decay, J. Fac. Sci. Univ. Tokyo ${\bf 28}$ (1982), 415-437. MR 83k:10056

Similar Articles

Retrieve articles in Journal of the American Mathematical Society with MSC (1991): 11F55, 11F70, 11F72

Retrieve articles in all journals with MSC (1991): 11F55, 11F70, 11F72


Additional Information

Hervé Jacquet
Affiliation: Department of Mathematics, Columbia University, New York, New York 10027
Email: hj@math.columbia.edu

Erez Lapid
Affiliation: Department of Mathematics, Weizmann Institute, Rehovot, Israel
Email: erezl@wisdom.weizmann.ac.il

Jonathan Rogawski
Affiliation: Department of Mathematics, Hebrew University, Jerusalem, Israel
Address at time of publication: Department of Mathematics, University of California, Los Angeles, California 90095
Email: jonr@math.huji.ac.il

DOI: https://doi.org/10.1090/S0894-0347-99-00279-9
Received by editor(s): October 14, 1997
Received by editor(s) in revised form: May 14, 1998
Additional Notes: The first author was partially supported by NSF Grant 9619766.
The third author was partially supported by NSF Grant 9401466.
Article copyright: © Copyright 1999 American Mathematical Society

American Mathematical Society